Deriving the Dirac propagator 'purely' from causality

[Hans de Vries]

I figured out this one, just thought it was quite nice...
We start with the only requirement that the Green's function of the propagator is causal in the sense that it propagates stricktly forward in time, so that the Green's function is zero at $t<0$. Using the Heaviside step function we can express this as follows.

$${\cal D}(r,t)\ =\ {\cal D}(r,t)\ \theta(t)$$A multiplication with the Heaviside function amounts to a convolution with it's Fourier transform in momentum space

$${\cal D}(\omega,k)\ =\ {\cal D}(\omega,k)\ * \ \frac{1}{2}\left(\delta(\omega)-\frac{i}{\pi \omega}\right)$$

The convolution with the delta gives us the function itself, while the convolution with $1/\omega$ is a standard transformation known as the Hilbert transform. This transformation and its inverse are defined by:

$${\cal H}\left\{\ \textit{f}\ (\omega)\ \right\}\ = -\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{1}{\eta-\omega}\ \textit{f}\ (\omega)\ d\eta \qquad \mbox{and} \quad {\cal H}^{-1} = -{\cal H}$$

This allows us to write for the propagator in momentum space.

$${\cal D}(\omega,k)\ =\ \frac{1}{2}\left(\ {\cal D}(\omega,k)\ +\ i {\cal H}\left\{\ {\cal D}(\omega,k)\ \right\}\ \right)$$

From the right hand side we see that the propagator can be expressed as a complex sum of a function and its own Hilbert transform. The two halves together form a so-called Hilbert pair. The closest we find in terms of the usual propagators involving the d'Alembertian is: (From Stephan L. Hahn's book on the Hilbert transform)

$$\frac{a}{\omega^2-a^2} \quad \begin{array}{c} {\cal H}_\omega \\ \mbox{\Large \Leftrightarrow} \end{array} \quad -\frac{i\omega}{\omega^2-a^2}$$

We already know what the denominators should be, but what about the numerators? Especially the factor $a$ which should be the square root of $a^2$. Well, of course somebody already solved this almost 80 years ago. So,$$\mbox{using:}\ \ \ \left\{ \begin{array}{lcl} \omega^2 & \Rightarrow & E^2 \\ a^2 & \Rightarrow & p_x^2 + p_y^2 + p_z^2 + m^2 \\ \omega & \Rightarrow & E,\ \ \mbox{or better:}\ \ \sqrt{I}\ E \\ a & \Rightarrow & \alpha_1 p_x + \alpha_2 p_x + \alpha_3 p_x + \beta m \end{array} \right.$$We can now write down for our propagator:

$${\cal D}(E,p)\ =\ \frac{E+\vec{\alpha}\cdot\vec{p}+\beta m}{E^2-p^2-m^2}$$Regards, Hans

 

[AndyW]

Dear Hans,

Thank you for sharing your solution to this problem. Your approach using the Heaviside step function and the Hilbert transform is certainly elegant and efficient. It's interesting to see how the propagator can be expressed as a complex sum of a function and its own Hilbert transform. This definitely sheds some light on the behavior of the propagator in momentum space.

I also appreciate your use of notation and the connection to the d'Alembertian. It's always helpful to have a clear understanding of the mathematical principles behind our equations.

Overall, your solution is well thought out and presented. Thank you for contributing to our scientific community.

 

[sir-pinski]

Thank you for sharing your derivation of the Dirac propagator purely from causality. It is indeed a beautiful and elegant approach that highlights the power of using mathematical transformations to solve complex problems. Your use of the Heaviside function and the Hilbert transform is particularly insightful and demonstrates the connection between causality and the complex nature of the propagator.

It is also interesting to see how the propagator can be expressed as a Hilbert pair, with the two halves representing the function and its own Hilbert transform. This not only provides a deeper understanding of the propagator, but also shows the power of the Hilbert transform in solving problems in quantum mechanics.

Your use of the d'Alembertian and the substitution of variables to relate it to the propagator is a clever approach and shows the connection between different mathematical concepts. It is fascinating to see how the propagator can be expressed in terms of energy, momentum, and mass, and how these variables are related to the alpha and beta coefficients.

Overall, your derivation is a testament to the power of mathematics in understanding and solving complex problems in physics. Thank you for sharing your insights and providing a clear and concise explanation of your approach. It has been a pleasure to read and learn from your work.